Steady-State Supersaturation Distributions for Clouds under Turbulent Forcing

Manuel Santos Gutiérrez aDepartment of Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot, Israel

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Kalli Furtado bCentre for Climate Research Singapore, Meteorological Service Singapore, Singapore

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Abstract

The supersaturation equation for a vertically moving adiabatic cloud parcel is analyzed. The effects of turbulent updrafts are incorporated in the shape of a stochastic Lagrangian model, with spatial and time correlations expressed in terms of turbulent kinetic energy. Using the Fokker–Planck equation, the steady-state probability distributions of supersaturation are analytically computed for a number of approximations involving the time-scale separation between updraft fluctuations and phase relaxation, and droplet or ice particle size fluctuations. While the analytical results are presented in general for single-phase clouds, the calculated distributions are used to compute mixed-phase cloud properties—mixed fraction and mean liquid water content in an initially icy cloud—and are argued to be useful for generalizing and constructing new parameterization schemes.

Significance Statement

Supersaturation is the fuel for the development of clouds in the atmosphere. In this paper, our goal is to better understand the supersaturation budget of clouds embedded in a turbulent environment by analyzing the basic equations of cloud microphysics. It is found that the turbulent characteristics of an air parcel substantially affect the cloud’s supersaturation budget and hence its life cycle. This is also shown in the context of mixed-phase clouds where, depending on the turbulent regime, different liquid-to-ice ratios are found. Consequently, the theoretical approach of this paper is crucial to develop tools to parameterize small-scale atmospheric features, like clouds, into global circulation models to improve climate projections for the future.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Manuel Santos Gutiérrez, manuel.santos-gutierrez@weizmann.ac.il

Abstract

The supersaturation equation for a vertically moving adiabatic cloud parcel is analyzed. The effects of turbulent updrafts are incorporated in the shape of a stochastic Lagrangian model, with spatial and time correlations expressed in terms of turbulent kinetic energy. Using the Fokker–Planck equation, the steady-state probability distributions of supersaturation are analytically computed for a number of approximations involving the time-scale separation between updraft fluctuations and phase relaxation, and droplet or ice particle size fluctuations. While the analytical results are presented in general for single-phase clouds, the calculated distributions are used to compute mixed-phase cloud properties—mixed fraction and mean liquid water content in an initially icy cloud—and are argued to be useful for generalizing and constructing new parameterization schemes.

Significance Statement

Supersaturation is the fuel for the development of clouds in the atmosphere. In this paper, our goal is to better understand the supersaturation budget of clouds embedded in a turbulent environment by analyzing the basic equations of cloud microphysics. It is found that the turbulent characteristics of an air parcel substantially affect the cloud’s supersaturation budget and hence its life cycle. This is also shown in the context of mixed-phase clouds where, depending on the turbulent regime, different liquid-to-ice ratios are found. Consequently, the theoretical approach of this paper is crucial to develop tools to parameterize small-scale atmospheric features, like clouds, into global circulation models to improve climate projections for the future.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Manuel Santos Gutiérrez, manuel.santos-gutierrez@weizmann.ac.il

1. Introduction

Water vapor supersaturation is a key thermodynamic parameter in the formation and development of warm and cold clouds. The activation of cloud condensation nuclei (CCN), the spectrum of droplet sizes, and therefore, precipitation heavily depend on the supersaturation budget of an evolving cloud. By extension, the study of supersaturation is fundamental to determine the radiative properties of cloud fields which constitute a major climate feedback (Khain and Pinsky 2018), yet a major source of uncertainty in climate projections (Zelinka et al. 2020).

Due to its microphysical character, supersaturation cannot be explicitly resolved by global circulation models (GCMs), and hence, it has to be prescribed using adequate parameterization schemes. One way of doing this is by predicting the probability of encountering a cloud parcel at a certain relative humidity level in terms of the background dynamics and large-scale information. Taking the GCM perspective, the idea is to infer the subgrid cloud properties out of the prognosed variables: vertical velocity, temperature, specific humidity, and pressure, which altogether depict the relative humidity configuration of a homogeneous cloud parcel. However, localized temperature gradients and turbulence at the smaller scales can create inhomogeneous fluctuations which alter the spread of supersaturation values, hence changing the microphysical properties of the subgrid cloud parcel.

Turbulence, in fact, influences the life cycle of clouds, from the smallest warm cumuli to the coldest stratiform clouds (Shaw 2003). Indeed, the incorporation of turbulent models into the determination of microphysical properties has played a key role in explaining the broadening of droplet size spectra in developing cumulus clouds (Grabowski and Abade 2017; Abade et al. 2018) and mixed-phase clouds (Hoffmann 2020; Chen et al. 2023), as well as in the activation and maintenance of long-lasting supercooled liquid water in icy clouds (Korolev and Field 2008; Hoffmann 2020). Moreover, laboratory experiments suggest that turbulence at the smallest scales are also responsible for CCN activation even in mean-subsaturated environments (Prabhakaran et al. 2020).

It is crucial, then, to determine not only the mean value of supersaturation at a given location, but also its variance or even higher moments. To this end, the equations for supersaturation or condensational growth are coupled with a suitable stochastic forcing law that captures the effects of turbulence affecting both vertical velocity fluctuations (Bartlett and Jonas 1972; Sardina et al. 2015) as well as the integral radius of the droplet distribution (Manton 1979; Cooper 1989). Hence, the classical deterministic models become stochastic differential equations (SDEs), which have been widely used in the cloud physics community; see, e.g., McGraw and Liu (2006), Paoli and Shariff (2009), Sardina et al. (2015), and Siebert and Shaw (2017). In fact, this stochastic physics framework has been employed in the elaboration of analytically tractable parameterization schemes for mixed-phase clouds (Furtado et al. 2016), the study of droplet growth by condensation (Bartlett and Jonas 1972; Sardina et al. 2015), and the determination of steady-state warm cloud properties (Siewert et al. 2017).

The supersaturation equation, also known as Squires equation (Squires 1952), consists of a nonlinear relation between supersaturation with its sources and sinks, and has extensively been studied; see, e.g., Korolev and Mazin (2003) or Devenish et al. (2016)—we also refer to Eq. (2) of this paper. The addition of turbulent updrafts modeled by stochastic processes into the linearized Squires equation allowed to obtain Gaussian formulas for the ice-supersaturation (probability) distribution (Field et al. 2014), which serves to diagnose the amount of supercooled liquid water in mixed-phase clouds (Furtado et al. 2016). Along these lines, Sardina et al. (2015) also provided a relation for the moments of supersaturation and droplet size distribution. However, some restrictions were imposed in these works. First, the full, nonlinear form of Squires equation has never been examined in the stochastic context, unlike its deterministic counterpart [Korolev and Mazin 2003, Eqs. (9) and (10)]. In fact, this nonlinear form is specially relevant for clouds experiencing high maximum supersaturation, as pointed out in Devenish et al. (2016). Second, in the limit of fast turbulent decorrelation—relative to supersaturation time scales—the stochastic turbulence becomes white noise, which can be a strong assumption if one is concerned with other shorter relevant time scales like that at the CCN activation level, as already warned by Field et al. (2014) and Abade et al. (2018). Third, the time invariance of mean droplet radius—the quasi-steady assumption (Squires 1952)—appears critical to obtain analytical supersaturation distributions (Field et al. 2014; Furtado et al. 2016). Turbulence, mixing, and entrainment, however, can provoke fluctuations in droplet integral radius (Manton 1979; Cooper 1989), which consequently perturb the supersaturation budget. This work is concerned with analytically assessing and lifting the listed assumptions in the study of the supersaturation equation.

This paper is structured as follows. In section 2, the Squires equation for the evolution of supersaturation is derived, and second, a stochastic equation for turbulent updrafts is presented, in the lines of the theory of stochastic Lagrangian turbulent models; see, e.g., the work of Rodean (1996). In section 3, the quasi-steady equation—which assumes a constant mean droplet/particle radius—is analyzed in a number of approximations providing new formulas for the probability distribution of supersaturation. In section 4, fluctuations in droplet size are allowed, and their net effects on supersaturation distribution are investigated in analytical terms. A total of five different supersaturation distributions are derived. In section 5, the relevance of the five obtained probability density functions is discussed and compared in the context of mixed-phase clouds: we compare the supercooled liquid cloud fraction and liquid water contents predicted by each distribution. Finally, a discussion over the results is done in section 6. To supplement the information in the main text, appendixes are included to discuss some technical topics related to the analysis of stochastic differential equations.

2. The supersaturation equation

a. The Squires equation

We consider a vertically moving cloud parcel containing a monodisperse family of liquid water droplets or ice particles—allowing mixed-phase conditions—that are spatially uniformly distributed. Furthermore, it is assumed that the number of droplets/particles per unit mass is constant in time. The evolution of supersaturation in the cloud is, essentially, determined by the sources and sinks of relative humidity due to the adiabatic cooling of the parcel in ascent, and condensation of water vapor onto the existing droplets’ or particles’ surface (Rogers and Yau 1989). However, the exact relation between the rate of change of supersaturation and its sinks and sources is obtained by taking its derivative with respect to time.

We recall that supersaturation—for either ice or water—is defined as
S=eEE,
where e is the water vapor pressure and E is the same, although at saturation over a flat surface of liquid water or ice. We shall not specify now whether we are dealing with water or ice supersaturation because the stochastic analysis will be done independently. It is noted, however, that the calculations immediately below can be done for single or mixed-phase clouds; see Korolev and Mazin (2003).
Taking the time derivative of S and employing the mass conservation, the Clausius–Clapeyron equation, and the quasi-hydrostatic approximation, Squires (1952) derived an equation to describe the evolution of the supersaturation budget—see also the appendix in the paper of Korolev and Mazin (2003):
11+SdSdt=awbdqdt,
where w and q are the vertical velocity and liquid-water or ice mixing ratio, respectively. The parameters a and b are the adiabatic and microphysical constant: see appendix A for the precise definition. The evolution of S obeys an equation with a nonlinear term—b(1 + S)dq/dt—stemming from the time and temperature dependence of the equilibrium water vapor pressure E in Eq. (1). Equation (2) reveals that only to leading order in small values of S do we obtain a linear dependence on vertical velocity w and water vapor condensation/deposition, dq/dt. Such is the case of warm clouds, where supersaturation levels do not typically exceed a few percent (Prabha et al. 2011). However, mixed-phase conditions in stratiform clouds arise precisely when ice supersaturation fluctuates spreading beyond small values (Korolev and Field 2008; Field et al. 2014), making the nonlinearity in Eq. (2) more relevant. This will be discussed in the next section.
To find a closed model for supersaturation, it is necessary to include the equation describing vapor condensation/deposition, dq/dt. The latter is described as [Korolev and Mazin 2003, Eq. (5)]
dqdt=4πNρa01ρρ+0h(r,ρ,c)ρr2drdtdrdρdc,
where h is the normalized particle size distribution and the different variables are understood for either liquid droplets or ice particles. The parameters N and ρa denote the particle concentration and density of air, respectively. It is worth noting that the first integral ranging from zero to unity refers to the possible distribution of capacitances c, which reflects the efficiency with which droplets/ice particles collect water vapor (Rogers and Yau 1989). While the capacitance of droplets is approximately 1—since all of them are almost spheres—ice particles possess a wider range of possible shapes that are displayed under certain temperature and humidity conditions, affecting the collection of water vapor. Also, a degree of freedom is left for the density of each particle, ρ, which again, for the case of ice phase, it can range between different values from particle to particle.
We now employ the diffusional growth law for droplets/particles, which states that water vapor condenses/deposits proportionally to the available supersaturation and inversely proportionally to the radius of the particle (Rogers and Yau 1989):
drdt=cASr,
where A is defined in the appendix A. Since we are considering a monodisperse particle size and shape, the previous equation does not only model the diffusional growth of a single particle but of the whole population. Inserting Eq. (4) into Eq. (3) we find
dqdt=S4πc¯ρ¯ANrρa,
where the overbars indicate mean properties of the ensemble of particles. Finally, to obtain the full evolution equation for S, the expression for water vapor absorption in Eq. (5) is substituted into Eq. (2). The updraft term is not yet specified, although we anticipate that an SDE for updrafts will be coupled to the supersaturation equation in section 2b.
It is clear, then, that a closed equation for the evolution of supersaturation can be obtained by simply integrating Eq. (5) and plugging into Eq. (2) to give
11+SdS(t)dt=bS(t)r(0)2+2cA0tS(s)ds+aw.
The solution of this equation has been shown to converge to a steady-state value, which is understood as an equilibrium supersaturation, whereby the changes in relative humidity are balanced by the absorption/release of water vapor by the droplets or ice particles (Korolev and Mazin 2003). Numerical solutions of this equation can be achieved, although its analytical treatment entails expansions and approximations (Devenish et al. 2016). Equation (6) is a nonlinear integrodifferential equation, since it possesses a square root of a memory or integral term 2cA0tS(s)ds which accounts for all past supersaturation configurations. The memory term is, in principle, only valid for all the time that droplets grow without fully evaporating, sedimenting, or precipitating. If any of these processes occurs, memory is lost and Eq. (6) would have to be reinitialized. To account for this issue, in section 4 we shall develop a theoretical framework to understand the net effects of the fluctuations in droplet or particle radius on the evolution of supersaturation.

b. Stochastic model for updrafts

The aim of this section is to construct a stochastic model for updraft fluctuations. For this, a cloudy parcel is supposed to be embedded in a turbulent environment with a prescribed surrounding supersaturation SE, which is allowed to be negative in case of dry, subsaturated air. The vertical velocity w in such cloud parcel is assumed to be decomposed into its mean w¯ and fluctuating part w′ so that w=w¯+w. Given the small spatial scales considered herein, turbulence is taken to be isotropic so that the variance of w′ is related to the turbulent kinetic energy (TKE) of the background flow in the following way:
TKE=12(σx2+σy2+σw2)=32σw2.
The updraft variance, σw2, together with the eddy dissipation rate ε provide an estimate of the exponential rate, 1/τd, at which the fluctuations w′ decorrelate in time. Such a number is called the Lagrangian decorrelation time scale (Rodean 1996). Roughly speaking τd > 0 indicates the amount of time needed for a turbulent flow to forget its original configuration:
τd=2σw2εC0,
where C0 is the Lagrangian structure-function constant and ε is the eddy dissipation rate. Hence, if we assume that vertical motion is homogeneous, random, with stationary mean w¯, variance σw2, and decorrelation time τd, the simplest model for the vertical velocity is the following red noise equation (Rodean 1996, section 3.5):
dw=1τd(ww¯)dt+2τdσwdWt,
where Wt denotes a standard Wiener process, which accounts for the acceleration increments over dt time units, that come from random pressure fluctuations which decorrelate instantly on time. Equation (9) has the structure of red noise or an Ornstein–Uhlenbeck process, for which there exists a vast collection of analytical results—see, e.g., Uhlenbeck and Ornstein (1930) or Pavliotis (2014). In particular, this one-dimensional Gaussian process satisfies the following mean, variance, and correlation properties:
E[w(t)]=et/τdw(0)+(1et/τd)w¯,
Var[w(t)]=σw2(1e2t/τd),
E[w(t)w(0)]=σw2et/τd.
where w(0) is the initial value of the vertical velocity. Thus, as t tends to infinity, the solution of Eq. (9) will distribute according to a Gaussian function with mean w¯ and variance σw2.
As a result of considering a cloud in contact with the exterior, turbulent motion will also be in charge of mixing mass at the cloud edges; see Eytan et al. (2022) for a recent discussion of cloud edges and transition zones in cumulus clouds in terms of adiabaticity of their components. This process is assumed to be acting at a constant rate and will drive the supersaturation at the cloud toward an equilibrium value SE. In other words, turbulent mixing rates determine the characteristic time τmix to homogenize a cloud volume with its surrounding reservoir (Baker et al. 1984; Khain and Pinsky 2018):
τmix=(L2ε)1/3,
where L is the characteristic length of the turbulent zone. Therefore, a more general form for the supersaturation equation is studied so that mixing at the cloud edges is also taken into account:
11+SdSdt=awbSr1τmix(SSE).
Finally, it is enough to couple this equation to Eq. (9) and the diffusional growth, Eq. (4), to obtain a closed set of equations for the time evolution of supersaturation. We note that other modeling approaches have been carried out to derive stochastic equations for supersaturation evolution. Indeed, in the work of Paoli and Shariff (2009), the authors propose Langevin equations for vapor and temperature fluctuations and yields a closed equation for supersaturation and droplet growth.

3. Quasi-steady model statistics

The diffusional growth rate is inversely proportional to the radius of the droplet or particle in question. Therefore, small droplets or particles grow faster compared to larger ones. In this sense, if Eq. (4) is initialized with a large radius r2(0), it is expected that it will remain almost constant, at least for the interval where the following inequality is satisfied; see also Korolev and Mazin (2003) and Devenish et al. (2016):
r2(0)|2cA°0tS(s)ds|.
Hence, assuming that changes in the size of the cloud droplets can be neglected, we can take r in Eq. (3) to be constant, equal to r¯. This is called the quasi-steady approximation (Squires 1952; Korolev and Mazin 2003). The resulting equation loses the time dependence of the variable r and reads as
dSdt=(B+C)(SCSEB+C)(1+S)+a(1+S)w,
dw=1τd(ww¯)dt+2τdσwdWt,
where two constants have been introduced:
B=bB0Nr¯;
C=(εL2)1/3.
Note that the radius r is no longer a variable and that is absorbed into the constant B and B0 is defined in the appendix A. The parameter B−1 is therefore the time scale associated with microphysical processes. The mixing time-scale parameter C−1 is here treated as independent with respect to TKE. We are therefore aiming at analyzing supersaturation statistics as a function of the different time scales involved in Eq. (14), so that the characteristic length scales L are susceptible of changing if different values of TKE are considered. The target of this section is to derive analytically the stationary statistics of model (14) in a variety of approximations that are detailed in the subsections bellow.

a. Fast Lagrangian decorrelation time scale

There are two different time scales involved in Eq. (14). One is set by the constant B + C, as the characteristic time for the consumption of supersaturation, and the other is the Lagrangian decorrelation time-scale τd. When the Lagrangian decorrelation time scale is small, the turbulent flow takes less time to forget its initial configuration compared to the typical time of approach to the equilibrium of supersaturation. This is, in the limit of (B + C)τd → 0, w(t) will become a stochastic delta-correlated process relative to the evolution of supersaturation; see also Sardina et al. (2015). Indeed, by referring to the theory of homogenization (Pavliotis and Stuart 2008, chapter 11, result 11.1), we are able to, mathematically rigorously, reduce the two-dimensional system describing S and w to
dS=[(B+C)(SCSEB+C)](1+S)dt+Aw¯(1+S)+A(1+S)dWt,
where a new constant A has been introduced and defined as
A=aσw2τd,
which is the normalization constant necessary to take this diffusion limit; see Rodean (1996, section 6.3). Because we have multiplicative noise—i.e., the state variable S multiplies the Wiener increment in Eq. (16): A(1 + S)dWt—we have to specify the stochastic calculus formalism being employed. For simplicity, Eq. (16) shall be studied under the Itô formalism, although the Stratonovich viewpoint can be taken instead (Gardiner 2009). For completeness, we note that in order to convert the Stratonovich version of Eq. (16) into Itô, we apply the Itô-to-Stratonovich correction H(S) which reads as (Gardiner 2009; Pavliotis 2014)
H(S)=A22S(1+S)2A22(1+S)S(1+S)=A22(1+S).
The resulting Itô stochastic differential equation is
dS={(B+C)[S2CSEA22(B+C)]}(1+S)dt+Aw¯(1+S)dt+A(1+S)dWt.
Notice that the equilibrium supersaturation SE is now modified by the term −A2, resulting from the mere presence of multiplicative noise. The best choice of stochastic formalism is not discussed here, although this equation reveals that the full form of the Squires equation encodes nonlinear interactions between supersaturation and turbulent fluctuations which yield nontrivial corrections to the stochastic formulation of supersaturation evolution.
Because the system is stochastic, the solutions of Eq. (16) give different results for each noise realization. To solve this problem, averages over all possible realizations are taken so the system is described in terms of probability distributions. For that, the Fokker–Planck equation describes the evolution of probability density functions associated with an SDE (Risken 1989; Saito et al. 2019), in this case Eq. (16):
tf(S,t)=S[(B+C)(SCSE+Aw¯B+C)(1+S)f(S,t)+A22S(1+S)2f(S,t)],
where f(S, t) indicates—when normalized—the probability of encountering a supersaturation of S at time t. Roughly speaking, this equation says that a density function is advected and diffused by the linear and stochastic components of Eq. (16), respectively.
In the spirit of Field et al. (2014), we shall assume that in a developing cloud the supersaturation budget quickly approaches stationarity, so that the probability of encountering certain value of supersaturation is provided by the stationary version of the Eq. (20), i.e., by setting (/t)f(S,t)=0. Unlike the analytical calculations of Field et al. (2014), the multiplicative noise and nonlinearities involved in Eq. (16) suggest that the resulting probability densities f cannot be Gaussian. Indeed, after successive integrations detailed in appendix B, the normalized time-independent density is
f1(S)=αα(β+1)1eαΓ[α(β+1)1]eαS(1+S)α(β+1)2,
where we have introduced the nondimensional parameters α, β, and S* defined as
α=2(B+C)A2;
β=αS*;
S*=CSE+Aw¯B+C.
The yielding moments can be computed accordingly in terms of successive gamma functions. The general formula for the uncentered moments of this distribution is
E[(1+S)n]=αnl=1n2[α(β+1)+l],
noting that
E[Sn]=E[(1+S)n]k=0n1(nk)E[Sk].
In particular, the formula for the variance is
Var(S)=E[(SE[S])2]=1α+(αβ1)2α2.
The parameter α is the nondimensional ratio of the turbulent and mixing time scales. Hence, it follows that a large value of α ≫ 1 yields a smaller supersaturation variance. Moreover, in the limit of large α, the function f1 can be recast into a Gaussian by means of Laplace’s method; see, e.g., Butler (2007).
As t tends to infinity, the expected value of S will converge to an equilibrium value at a characteristic rate given by the phase relaxation τp (Korolev and Mazin 2003). If the dynamics were deterministic, such rate would be given by
τp=1Aw¯+B+C,
which is obtained by examining the exponent of the solution of the ordinary differential Eq. (19); see details in Korolev and Mazin (2003). In the stochastic context, such relaxation rate is obtained by taking the expectation over all noise realizations. However, it would be necessary to solve an open system involving higher-order moments of supersaturation; cf. Sardina et al. (2015) or Siewert et al. (2017). Hence, it is not immediately clear whether Eq. (25) is still the phase relaxation for noisy updraft fluctuations. This is numerically checked for a cold, icy cloud embedded in a turbulent environment in Fig. 1. The solution of the deterministic version of Eq. (19) is solved and compared, in Fig. 1, against an ensemble mean of Eq. (19), both cases subject to a mean updraft of w¯=0.2ms1. Such ensemble mean is calculated by taking 103 noise realizations and averaging them at every time step. The procedure is repeated 10 times to obtain the mean (blue line) and standard deviation (gray shade). We also include the relaxation curve for the linear approximation that will be explained in the next section. The initial condition of supersaturation is taken to be 0.5 for demonstration purposes. It is observed that the deterministic relaxation (orange curve) provides a good estimate for the stochastic one (blue curve).
Fig. 1.
Fig. 1.

Phase relaxation to equilibrium. The blue curve represents the average of 10 ensemble means of 103 noise realizations of Eq. (19) for an icy cloud, with a prescribed mean updraft of w¯=0.2ms1. The gray shading shows plus and minus two standard deviations of the 10 ensembles. The orange curve is the solution of Eq. (19), where noise is vanishing. The green curve is the exponential relaxation time obtained from the linear Eq. (26). The ambient conditions in this numerical experiment are S(0) = 0.5, T = −10°C, p = 50 500 Pa, r¯=5μm, and Ni = 100 L−1.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

Linear approximation

For warm-phase stratocumulus and cumulus clouds, supersaturation does not go beyond 1%, so that the approximation 1 + S ≈ 1 is widely taken in the literature as a first-order approximation of the Squires equation (Pruppacher and Klett 2010; Khain and Pinsky 2018). Under this approximation and some algebraic manipulations, Eq. (16) becomes an OU process—which was already mentioned earlier around Eq. (9):
dS=(B+C)(SS*)dt+AdWt.
Such linear stochastic differential equation describes a process which has an invariant density equal to a Gaussian distribution. In particular, mean and variance are given by
E[S(t)]=e(B+C)tS(0)+S*[1e(B+C)t],
Var[S(t)]=1α[1e2(B+C)t]=A22(B+C)[1e2(B+C)t].
As t tends to infinity, the supersaturation value subject to a constant mean updraft tend to an equilibrium value S* exponentially fast, with rate given by (B + C). Hence, the steady-state statistics are provided by a Gaussian distribution:
f2(S)=α2πeα(SS*)2/2.
Unlike the nonlinear supersaturation equation investigated in section 3a, the phase relaxation for the present linear version yields the same relaxation time as its deterministic analog. This is given by
τp=1B+C.
Note that the phase relaxation in Eq. (25) is different to Eq. (29), in that the mean updraft w¯ is not present in the latter. Consequently, it is only in the limit of w¯0 when both equations converge to equilibrium at the same rate. For illustration, Eq. (27a) is plotted in Fig. 1 in green color, and demonstrates that the nonlinearity accelerates the convergence to equilibrium.

b. Slow Lagrangian decorrelation time scale

When there is no time-scale separation between the supersaturation evolution and the updraft fluctuations, the limit of τd/τp going to zero cannot be taken and therefore the white noise approximation of the previous section is not valid, as already pointed out in Field et al. (2014) and Abade et al. (2018). For instance, if number concentration increases—thus reducing τp—the white noise approximation does not hold any more. Here we provide a formula for the distribution of supersaturation in the presence of turbulent updraft fluctuations with exponentially decaying correlation rate τd1. Unfortunately, the resulting nonlinear Eq. (14) possesses statistics that are intractable analytically. Hence, we start from Eq. (12) and assume, as in section 1, that 1 + S ≈ 1. We obtain the following two-dimensional stochastic linear equation:
{dS=(B+C)(SCSEB+C)dt+awdt,(30a)dw=1τd(ww¯)dt+2τdσwdWt,(30b)
which can be compactly recast into matrix form:
d[Sw]=B([Sw]m)dt+AdWt,
where Wt is a two-dimensional independent Wiener process and where the matrices B and A and the vector m are defined as
B=[b11b120b22]=[(B+C)a01/τd],
A=[000a22]=[000(2/τd)1/2σw],
m=B1[CSEw¯τd]=[CSE+aw¯B+Cw¯].
Note that the Eq. (30) is degenerate—i.e., noise affects directly to only one variable—although it will possess an invariant distribution with a smooth density function. However, the two-dimensional covariance matrix cannot be obtained straightforwardly out of the noise law, but by computing the following matrix integral (Pavliotis 2014, proposition 3.5):
Σ=[σ11σ12σ12σ22]=0eBsAATeBTssds=[a2σw22(B+C)(B+C+1/τd)aσw22(B+C+1/τd)aσw22(B+C+1/τd)σw2],
where the symbol “T” denotes the transpose vector/matrix. The resulting process possesses, as in the one-dimensional case, a Gaussian stationary distribution where now
f3(S,w)=(2π)1(detΣ)1/2e1/2([S,w]mT)Σ1([S,w]Tm).
To derive the marginal distribution for S we apply the affine projection P = [1, 0] to the random variable [S, w]T so that P[S, w]T = S. Hence,
P[Sw]=SN(Pm,PΣPT)=N[CSE+aw¯B+C,a2σw22(B+C)(B+C+1/τd)].
The time-scale separation argument to reduce the equation for the evolution of supersaturation is only valid for small values of TKE which yield small Lagrangian decorrelation time-scales τd relative to phase relaxation time τp. This hypothesis is tested and shown in Fig. 2, where the variance of ice supersaturation in an icy cloud is computed using the formulas here presented and its numerical estimation using long time series of 106 seconds. The equations are integrated using a simple Euler–Maruyama method with a time step of 10−2 s (Gardiner 2009). It is noted that changing TKE implies a change in the characteristic length scale of the turbulent zone L—here on the order of 1000 m—since the mixing time-scale C−1 is kept constant. Consequently, this figure shows the dependence of ice-supersaturation variance as a function of updraft fluctuation time scales where the homogenization time, namely, C−1, is the same.
Fig. 2.
Fig. 2.

Supersaturation variance as a function of TKE. The blue colors correspond to the variances for the 2D linear system (30) with red-noise updrafts, whereas the black colors correspond to the homogenized 1D Eq. (26), where updrafts become white in time. The dots are obtained by calculating the variance of 106-s time series with a time step of 10−2 s for each value of TKE. The solid lines are the predicted variance using Eqs. (22a) and (35), for the blue and orange curves, respectively. We highlight that when turbulence is less energetic, both estimates become more similar. The ambient conditions in this numerical experiment are SE = 0, w¯=0, T = −10°C, p = 50 500 Pa, r¯=5μm, and Ni = 100 L−1.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

4. Diffusional growth and size fluctuations

In the section 2 we presented the full and closed equation for the evolution of supersaturation accounting for the condensational (depositional) growth of droplets (ice crystals) coupled to turbulent updraft fluctuations; see Eq. (6). In section 3, we investigated the properties of the stochastic Squires equation, describing the evolution of supersaturation in the quasi-steady approximation, where condensational growth is neglected. Here, we aim at lifting the quasi-steady assumption, by coupling supersaturation to condensational growth. In this regard, analytical work is limited because of the complexity of the resulting equations (Korolev and Mazin 2003). Nevertheless, approximations have been made to calculate the solutions of the supersaturation equation for small times (Devenish et al. 2016), and to determine the linear growth of variance of droplet radius as time increases (Sardina et al. 2015). In this section we shall provide, first, a brief review of the problem of droplet radius variance increase under random updrafts and, second, present a modified supersaturation equation that accounts for random fluctuations in droplet radius. The novel results are two different probability distributions for supersaturation in the spirit of the previous sections.

To develop our analysis, two default approximations are taken in this section. First, we will assume that updraft fluctuations decorrelate instantly so that the white-noise model is valid. Second, the small-supersaturation approximation is taken: 1 + S ≈ 1. Then, by applying the chain rule to Eq. (4), we can write the supersaturation equation coupled to diffusional growth as
{dS=C(SSd*)dtBdrSdt+AdWt(36a)dr2=FSdt,ifr20andS0(36b)dr2=0,ifr2=0andS<0,(36c)
where some constants are introduced:
Sd*=CSE+Aw¯C;
Bd=bB0N;
F=2cA.
We note that the coupling of Eqs. (36a) and (36b) is nonlinear since it involves the square root of the variable r2 and makes the analytical work intractable. However, we refer at this stage to appendix C for a note on the analysis of the square root stochastic process.
When the nonlinear term rS is small and close to zero, the mean and variance of r2 evolve according to
E[r2(t)]=FtE[S(s)]ds+r2(0)FeCtC[Sd*S(0)]+FSd*t+r2(0),
E{[r2(t)]2}=F2E[0t0tS(s)S(u)dsdu]=2F20t0uA22C[eC(us)eC(u+s)]dsdu,
=F2A2C2tF2A22C3+F2A22C3(4eCte2Ct).
These set of formulas are identical to those obtained in the early twentieth century in the study of Brownian motion—see Einstein (1905) and Uhlenbeck and Ornstein (1930)—and imply that the evolution of the squared radius follows Brownian paths where, in particular, the mean-squared displacement scales linearly for large times, if negative values of squared radii were allowed. Indeed, if t ≫ 1 and the nonlinear coupling is small (Sardina et al. 2015),
E{[r2(t)]2}F2A2C2t.
The boundary condition r2 = 0 is strictly necessary since it is possible that trajectories of r(t) in Eq. (36b) vanish. When that happens, it means that the particles or droplets in question have evaporated and that the formula for the variance in Eq. (38c) has to be reinitialized once the droplets and particles have reactivated.

While under this framework there is no stationary distribution with finite variance for particle radius, it was shown in Siewert et al. (2017) that if the ambient supersaturation SE is negative, i.e., subsaturated, the probability distribution of r2 will possess the structure of an exponential function with a Dirac peak located at r2 = 0, which arises from the boundary condition in Eq. (36c). Indeed, it is expected that the trajectories in the Sr2 plane of Eq. (36) will display cycles, where r2 grows but then vanishes and sticks at the boundary of r2 = 0 for an open interval of time; see Fig. 4 of Siewert et al. (2017).

a. Fluctuations in droplet radius

In the previous section we clarified that if supersaturation is let to be driven by random turbulent updrafts, the mean-square radius grows linearly in time, and therefore, unbounded Brownian excursions can be expected when solving the condensational growth equation. This implies that variance is not bounded and steady-state distributions of droplet radius have infinite variance. In this section, the target is to compromise between the quasi-steady approximations of section 3 and the coupling with diffusional growth as done in Eq. (36). This is done by accounting for variability in integral radius due to mixing with the cloud’s exterior, perturbations in the particles’ capacitance—particularly for ice particles; see Eq. (4)—and small thermal fluctuations that affect the equilibrium vapor pressure at each droplet’s surface. Small radius fluctuations were already theoretically suggested in the work of Manton (1979) and further explored in Cooper (1989), where the authors consider their effect in the broadening of the droplet size spectrum. Under the quasi-steady approximation, it is concluded that updraft fluctuations alone cannot provoke a broadening of droplet size spectra, but turbulent fluctuations in the integral radius have to be considered; see Eq. (10) in Cooper (1989) and associated comments.

We shall therefore take the same strategy of Cooper (1989) and introduce the value σr, which represents the standard deviation of the random fluctuations of the radius variable r. To simplify the analytical expressions in this section, we shall present the results for SE = 0 and w¯=0. The starting point is the linear approximation—i.e., 1 + S ≈ 1—that together with white-noise fluctuations in droplet size lead to a new equation for supersaturation:
dS(t)=CS(t)dtBdS(t)[r¯dt+σrdWt(1)]+AdWt(2),
where W(1) and W(2) are two independent Wiener processes. Also, the parameter B in Eq. (15a) has been modified to Bd = bB0N. The first fluctuation term σrdWt(1) represent random fluctuations in mean radius, although they can also represent fluctuations in integral radius Nr¯ if the concentration parameter N is factorized out of Bd. The second fluctuation term AdWt(2) represents the same updraft fluctuations as investigated in sections 2 and 3. In this case, solving the stationary Fokker–Planck equation for S*=0 yields the following nonnormalized stationary distribution:
f4(S)=(A2+σr2Bd2S2)1(C+Bdr¯)/(Bd2σr2).
This formula is similar to a Cauchy-like distribution, for which higher momenta might not be well defined. However, by examining the exponent, it follows that the distribution f4 has M momenta, if M/2<1+(C+Bdr¯)/(Bd2σr2). The function f4 describes the steady-state statistics when Wt(1)Wt(2), but since fluctuations in droplets size typically originate in fluctuations of supersaturation, it is important to consider the case with Wt(1)=Wt(2), so that noise sources are correlated. In such scenario, the stationary distribution is
f5(S)=e[2(C+Bdr¯)A/(Bd2σr2)](ABdσrS)1(ABdσrS)2[2(C+Bdr¯)/(Bd2σr2)].
The probability distributions of Eqs. (41) and (42) differ from a Gaussian distribution since the noise appears in a multiplicative way. It is clear, though, that the stochastic process Eq. (40) will converge to linear evolution of supersaturation as σr tends to zero, which yields the Gaussian distribution of Eq. (28). It is possible, on the other hand, to show the pointwise convergence of Eqs. (41) and (42) to Eq. (28), by means of Laplace’s method; see, e.g., Butler (2007).

b. Supersaturation distribution comparison

At this stage we overview the qualitative mathematical features of the calculated supersaturation distributions {fk}k=15. A preliminary comparison is provided in Fig. 3, where all the PDFs for ice supersaturation are plotted for the same atmospheric conditions. The PDFs f1, f2, and f3 are those concerning the quasi-steady approximation and they all present distinct characteristics. The positive skew in f1 predicted by Eq. (23) becomes apparent and highlights the importance of considering the full, nonlinear Squires equation for small values of the nondimensional parameter α, as already pointed out in section 3. The time-scale separation between τd and τp is crucial, as is observed in the curves for f2 and f3 where the latter possesses a relatively more peaked distribution. Upon the introduction of radius fluctuations, two distributions are presented, f4 and f5. The most characteristic feature of the last two is that they possess thicker tails, as results from the addition of an extra fluctuating term. Indeed, this is more pronounced in Fig. 3b, where a nonexponential tail decay is observed in f4 and f5. For uncorrelated perturbations in updrafts and particle radius—corresponding to W1(1)Wt(2) in Eq. (40)—supersaturation spreads symmetrically on positive and negative values, while for f5—corresponding Wt(1)=Wt(2) in Eq. (40)—has a heavy negative skew. We recall that in the limit of vanishing σr, f4, and f5 would converge to f2.

Fig. 3.
Fig. 3.

Comparison between PDFs. The legend indicates the analytically obtained PDFs {fk}k=15. The example ambient conditions for the calculations are SE = 0, w¯=0, T = −5°C, p = 50 500 Pa, r¯=5μm, σr2=106m2, TKE = 5 m2 s−2, and Ni = 100 L−1.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

5. Calculating mixed-phase cloud properties

Mixed-phase clouds constitute a large portion of the global cloud coverage. Altocumulus, altostratus, stratocumulus, and Arctic stratus are instances where mixed-phase conditions have been observed to exist and persist at temperatures as low as −40°C; see, e.g., Korolev et al. (2017). The coexistence of liquid water and ice is, however, thermodynamically unstable so that, in freezing temperatures, ice will inevitably grow at the expense of liquid water (Bergeron 1935). It was suggested in Korolev and Field (2008) a mechanism by which supercooled liquid water can be activated in an initially icy cloud. According to such work, liquid water is activated in an icy cloud if the following two criteria are met: (i) the vertical velocity must exceed a threshold value and (ii) the cloud parcel must be lifted to a threshold altitude. As a consequence, dynamical forcing is responsible for the production of supercooled liquid water and, moreover, different updraft profiles yield different mixed-phase conditions.

It was already advanced in Korolev and Field (2008) that turbulent updrafts could be a mechanism to produced mixed-phase conditions faithful to what is observed, for example, in stratiform clouds; see also Li et al. (2019) for a study on the effects of supersaturation fluctuations on droplet growth on such clouds. Indeed, supersaturation fluctuations can provoke liquid-water saturation conditions in icy parcels, which translates to the activation of supercooled liquid water. In addition, numerical simulations indicate that small-scale turbulent mixing decelerates the ice growth and, thus, extends the lifetime of supercooled water (Hoffmann 2020). In Field et al. (2014), an analytical framework was established for determining mixed-phase properties for icy clouds in turbulent environments. Such framework is based on the analysis of the (ice) supersaturation equation subject to the linear approximation and random turbulent updrafts modeled by white noise; this is revisited in section 3a(1).

In this section, we shall investigate the implications of using the different turbulent models here proposed. The analytical supersaturation distributions {fk}k=15 are employed to calculate the supercooled liquid cloud fraction and mean liquid-water content (LWC) for an initially icy cloud as a function of the strength of the turbulent parameters: TKE and droplet radius fluctuations. We first recall the procedure described in Field et al. (2014) to convert ice-supersaturation Si to liquid water given a probability distribution for Si. Supersaturation with respect to ice relates to that of water, Sw, in terms of the ratio of their respective equilibrium vapor pressures, Ew(T) for water and Ei(T) for ice:
Si=Ew(T)Ei(T)Sw+Ew(T)Ei(T)1=η(T)Sw+η(T)1,
where η(T) = Ew(T)/Ei(T). As a consequence, supersaturation with respect to ice at water saturation Siw is given by
Siw=η(T)1.
Hence, the fraction of cloud in mixed-phase conditions and mean LWC are determined by how supersaturation spreads beyond Siw. According to the present work, such spread is determined by the probability density functions {fk}k=15, which model different turbulent and atmospheric conditions; we refer back to sections 3 and 4. Concretely, the fraction of cloud parcel that is taken to be in mixed phase is the total time Si spends above Siw or, in other words—by invoking ergodicity (Pavliotis 2014, chapter 1)—the integral of fk from Siw to infinity:
Cfk=limt1t0t1SiSiw[Si(s)]ds=Siwfk(Si)dSi,
where 1SiSiw is the characteristic function for values of Si larger than Siw. As noted in Field et al. (2014), in the Gaussian case the supercooled liquid cloud fraction is given explicitly by
Cf2=12erfc[α2(SiwS*)],
where erfc(⋅) is the error function of a standardized Gaussian distribution. Moreover, the parcel’s LWC is calculated by assuming that ice supersaturation above that with respect to liquid water is converted into droplets. The LWC is, then, estimated by integrating the adjustment formula of Eq. (43) to obtain the condensed water amount ⟨qk, for each k:
qk=limt1t0t1SiSiw[Si(s)][Si(s)Siw]ρaqsids=Siw(SiSiw)fk(Si)ρaqsidSi,
where ρaqsi is the factor that converts liquid water content to supersaturation values.

The five probability distributions obtained in the previous sections are now used to compute the supercooled liquid cloud fraction and the mean LWC of an adiabatic cloud parcel for a range of free parameters. Such free parameters are the TKE, as a proxy for turbulent forcing, and variance of the droplet radius fluctuations. In Fig. 4 we show the dependence of the mentioned partial moments on values of TKE at two temperatures indicated in the captions: Figs. 4a and 4b show supercooled liquid cloud fraction and mean LWC at −10°C, whereas Figs. 4c and 4d show the same at −5°C. Such statistics where computed using a simple quadrature scheme on the interval [−10, 10] so that all the considered PDFs integrate to unity with a tolerance of 10−10. We observe a monotone dependence on TKE in all the PDFs but for the gamma distribution, which yielded decreasing cloud fractions for values of TKE > 6 m2 s−2 at −10°C and TKE > 4 m2 s−2 at −5°C. Such change in trend is due to the displacement of the mode and tail thickness in the gamma distribution as the location factor is altered due to the multiplicative noise. It is noted that the red and orange curves, for f2 and f4, respectively, are almost coinciding.

Fig. 4.
Fig. 4.

(a),(c) Supercooled liquid cloud fraction and (b),(d) mean LWC as a function of TKE. The supercooled liquid cloud fraction and mean LWC are calculated using Eqs. (45) and (47), respectively, against TKE for each analytical supersaturation distribution {fk}k=15 and for two temperature configurations. The red and orange curves, for f2 and f4, respectively, are almost coinciding. In (a) and (b), a temperature of −10°C is considered, while in (c) and (d), a temperature of −5°C is considered. The rest of the ambient conditions are SE = 0, w¯=0, σr2=106m2, p = 50 500 Pa, r¯=5μm, and Ni = 100 L−1.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

When droplet size fluctuations are allowed to vary, the first three calculated distributions f1, f2, and f3 naturally yield the same statistics for supercooled liquid cloud fraction and mean LWC. Contrarily, when noisy variations in radius are allowed the distributions of f4 and f5 are likely to display a dependence on σr. This dependence is shown in Fig. 5, where the supercooled liquid cloud fraction and LWC are calculated as a function σr, for f2, f4, and f5. Figures 5a and 5b show the results at −10°C, whereas Figs. 5c and 5d show the same at −5°C. Because f4 and f5 are expensive to evaluate at small values of σr, ergodic averages are computed instead, i.e., the first equality in Eqs. (45) and (47). Such averages are taken over an integration of Eq. (40) over 106 seconds. While f2 is expectedly constant, f4 and f5 start to be dependent for values larger than σr21012m2, which corresponds to a standard deviation 20% of the mean droplet radius r¯. In particular, it is observed that Cf4 grows up to 1% when droplet fluctuations have a variance of 10−8 m2. On the other hand, Cf5 appears to be independent of σr. We recall that this case corresponds to when the fluctuations in r are independent of the noisy updraft. Such independence does not hold in LWC, where ⟨q5 appears to grow, as a consequence of the thickening of the tails.

Fig. 5.
Fig. 5.

(a),(c) Supercooled liquid cloud fraction and (b),(d) mean LWC as a function of σr. The supercooled liquid cloud fraction and mean LWC are calculated using Eqs. (45) and (47), respectively, against σr for the analytical distributions f1, f4, and f5 and for two temperature configurations. For small values of σr, the functions f4 and f5 are computationally expensive to evaluate so ergodic averages of 106 seconds are taken instead. In (a) and (b), a temperature of −10°C is considered, while in (b) and (d), a temperature of −5°C is considered. The rest of the ambient conditions are SE = 0, w¯=0, TKE = 2.612 m2 s−2, p = 50 500 Pa, r¯=5μm, and Ni = 100 L−1.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

6. Discussion

In this paper, the analysis of the stochastic Squires equation has been done, with the aim of providing analytical formulas for the distribution of supersaturation in an adiabatic cloud parcel. Such equation describes the evolution of supersaturation over time by, essentially, taking into account the sources and sinks of relative humidity due to (i) adiabatic cooling via turbulent updrafts and (ii) water vapor condensation onto droplets/particles. Turbulent effects on updrafts and droplet/particle radius are here modeled as a red-noise process, in accordance with the spatial and time correlations of isotropic fluctuations in the inertial regime (Rodean 1996). Thus, we presented a hierarchy of five supersaturation equations that, by using the Fokker–Planck equation, possess analytically tractable distributions whose properties were assessed in the context of mixed phase clouds.

The theoretical results of Field et al. (2014)—here condensed in section 3a(1)—have been here generalized to a wider range of contexts. First of all, the Squires equation is considered in section 3 in its full nonlinear version and has been shown to possess a gamma-like stationary distribution—here denoted as f1—that deviates from a Gaussian according to the parameter α defined in Eq. (22a). Such parameter, also found in Field et al. (2014), is a nondimensional ratio between the turbulent fluctuation time scales and the phase relaxation coefficient. Thus, as turbulent fluctuations decrease in variance (relative to the microphysical or mixing time scale), f1 becomes better and better approximated by the Gaussian distribution f2. Indeed, this can be seen in section 5 from the computation of partial moments for mixed-phase clouds—mixed cloud fraction and mean LWC—in Fig. 4.

The time-scale separation assumption between updraft fluctuations and supersaturation, needed to compute f1 and f2, is lifted in section 3b. Indeed, f1 and f2 are only valid when supersaturation time scales are much greater that the Lagrangian decorrelation τd so that updraft fluctuations become uncorrelated in time; cf. Pavliotis (2014, chapter 11, result 11.1). The resulting distribution f3 is still Gaussian, but possesses a modification in the variance compared to that of f2; see Eq. (35). In Fig. 2, we show that the variance predicted by f2 and f3 diverge as TKE increases.

The main assumption needed to compute f1, f2, and f3 is the quasi-steady approximation, whereby the droplet or ice particle radius is considered constant. When coupling droplet growth and supersaturation in a turbulent environment, it was shown in Sardina et al. (2015) that the long-term variance of droplet squared radius scales linearly in time, similar to Brownian motion. This is revisited here in section 4. We highlight that such result is only valid for short times, since large families of droplets are subject to processes like sedimentation, evaporation, or mixing with exterior dry air that provoke a memory loss in collective droplet growth, and hence, the variance formula is reinitialized. Here, we proposed a modification of the supersaturation equation where the sink term—which depends on the integral radius of the droplet/particle population—is allowed to fluctuate randomly, as suggested in earlier work (Manton 1979; Cooper 1989). Two situations are studied: (i) fluctuations in updraft are uncorrelated to those of droplet size and (ii) the source of noise is the same, albeit with different intensities. The resulting model yields analytically tractable probability distributions for supersaturation. The net effect of droplet radius fluctuations is illustrated in Fig. 5, where the supercooled liquid cloud fraction and mean LWC are computed as a function of σr. It is found that correlated radius and updraft fluctuations yield a probability distribution that deviates severely from the quasi-steady approximation by up to +5% in supercooled liquid cloud fraction when σr ≥ 10−6 m. Contrarily, the mean LWC is negatively correlated with fluctuations in σr in case of f4. Regarding the uncorrelated sources of noise, the supercooled liquid cloud fraction appears to be weakly dependent in σr. On the other hand, mean LWC correlates positively is droplet size fluctuation variance. All the obtained PDFs are also qualitatively compared in Fig. 3.

There is still a need to provide a more quantitative verification of the formulas here presented. However, the formulas and discussions in this paper suggest that the widely used linear approximation for supersaturation evolution—see Eq. (26)—is limited to clouds experiencing low supersaturation, where the quasi-steady approximation is valid and updrafts decorrelate instantly with respect to phase relaxation time scales. The mixed-phase cloud stochastic parameterization scheme developed in Furtado et al. (2016) is based on these assumptions, and hence, future work should be oriented toward discerning which specific atmospheric conditions are appropriate for each of the here calculated supersaturation PDFs {fk}k=15, in order to obtain better probabilistic formulas for the parameterization of subgrid cloud properties in GCMs.

On a more theoretical note, a deeper investigation of Eq. (6) would be of great interest in this stochastic framework. One step forward would be to impose a characteristic time for the loss of memory, so that the integrodifferential equation can be replaced by a simpler expression, possibly some form of noise with suitable time-decorrelation properties. We anticipate that this would entail technical difficulties due to the square root nonlinearity—here minimally tackled in appendix C—so research should be oriented in this direction. In general, we belief that this statistical-physics approach can be extended to more general contexts, possibly, by including more microphysical processes that affect the growth of liquid droplets or ice particles and, hence, the overall regulation of a cloud’s supersaturation budget.

Acknowledgments.

The authors thank S. Roncoroni, P. Field, and B. Devenish for their comments, suggestions, and kind reception at the Met Office. MSG is grateful to the Mathematics of Planet Earth Centre for Doctoral Training (MPE CDT) for making this collaboration possible. MSG acknowledges and is grateful for the support of the Institute of Mathematics and its Applications (Grant Number SGS21/08). MSG is thankful to I. Koren, M. D. Chekroun, the cloud physics group, and the graduate school at the Weizmann Institute of Science for providing a most inspiring environment.

Data availability statement.

No data have been used in this publication apart from the numerical integration of the equations here presented. The numerical schemes employed are found in https://doi.org/10.5281/zenodo.7904642.

APPENDIX A

List of Some Used Notations and Symbols

Table A1 provides a list of symbols that appear in the main text.

Table A1.

List of symbols.

Table A1.

APPENDIX B

Stationary Supersaturation Distribution

The calculation of the stationary distributions of each case study is done by studying the Fokker–Planck representation of the stochastic processes in question (Risken 1989; Saito et al. 2019). Such equation describes how probability distributions evolve in time toward its stationary state. For simplicity, in this appendix we will just show how to derive, step by step, the distribution for ice supersaturation Eq. (21). In this case, the Fokker–Planck equation associated with Eq. (16) is given by Eq. (20). Then, its stationary version, for SE=w¯=0, reads as
0=S{(B+C)S(1+S)f(S)+S[A22(1+S)2f(S)]},
where the time dependence has been dropped. Integrating from l (yet to be determined) to S and assuming that the stationary distribution f and (/S)f vanish at l:
0=(B+C)S(1+S)f(S)+A2(1+S)f(S)+A22(1+S)2Sf(S).
We now rearrange the equation to make it homogeneous on both sides:
Sf(S)f(S)=2(B+C)SA2(1+S)21+S.
Integrating on both sides,
log[f(S)]=2(B+C)A2log(1+S)2(B+C)A2S2log(1+S).
Taking exponentials on both sides,
f(S)=e[2(B+C)/A2]S(1+S)2+[2(B+C)/A2].
This is a nonnormalized solution for the stationary Fokker–Planck equation, which has a singularity at −1. As S tends to infinity, f(S) goes to zero asymptotically. We now calculate the normalization constant C, for which we employ the parameter α like in Eq. (22a):
C=1eα(1+S)(1+S)2+αdS=eαα1α0ezzα2dz=eαα1αΓ(α1).
Finally, f/C gives Eq. (21). This process is repeated for every probability distribution {fk}k=15, although the general formula is derived below.

The general case

A general one-dimensional SDE reads as
dx=V(x)dt+σ(x)dWt,
where Wt is a standard Wiener process and V and σ ≠ 0 have the regularity so that solutions distribute according to a smooth probability density function (Gardiner 2009; Pavliotis 2014). The associated Fokker–Planck equation is
tf=x{V(x)f+12x[σ2(x)f]},
where f is a probability density and a function of x and t. The stationary distribution of Eq. (B7) is obtained by setting tf = 0, and solving for f:
0=x{V(x)f+12x[σ2(x)f]}.
Integrating from l, where f is assumed to vanish, to x and rearranging to make the equation homogeneous,
xf(x)f(x)=2V(x)σ2(x)x[σ2(x)]σ2(x),
solving the indefinite integral on both sides yields
logf(x)=x2V(x)σ2(x)dxlog[σ2(x)],
where x denotes the indefinite integral. We now take the exponentials on both sides:
f(x)=ex2V(x)/σ2(x)dxσ2(x).
One is left with finding the normalizing constant.

APPENDIX C

The Square Root Process

The condensational growth equation is easily integrated into the supersaturation equation, although its expression depends on the square root of the initial particle size plus its fluctuating part due to supersaturation variations; see Eq. (6). If such fluctuations, here denoted as δS(t), decorrelate instantly and have zero mean, such expression is rewritten as
r(t)=r2(t)=r(0)2+2cA0tS(s)dsr(0)2+δS(t),
where the variance of δS(t) (σr22) is proportional to the diffusional constant A. Deriving the statistics of a squared root stochastic process is difficult, although in the limit of variance fluctuations of r2 begin small, or when the diffusional constant A is small, we can derive the steady-state mean and variance of the square root process as an expansion:
E[r]=E[r2]=r(0)18r(0)3σr22+O(σr24),
Var(r)=Var(r2)=E[r2]E[r2]2=14r(0)2σr22+O(σr24).
Surprisingly, the variance of the square root process only scales inversely proportionally to r(0)2. This approach assumes that r(0) is the mean radius and that fluctuations δS(t) are so small that r2 remains positive. This result is general and can be applied to any square root random variable, with a positive mean, and in the limit of small variance.

To support this analytical expansion, we numerically sampled an adimensional random variable X, normally distributed with mean 2 and standard deviation σX, where the latter takes 250 equispaced values between 10−4 and 0.5. The sample is of size 105 draws. With this set of data, we are able to numerically estimate the mean and variance of the square root random variable, X, discarding all samples that gave negative values. The aim is to predict the variance and mean of X, using the truncated expansions of Eq. (C2) and the prescribed values of σX. The truncation is done at O(σX4). Indeed, in Fig. C1 the analytical predictions—plotted in solid colored curves—match to a high degree of accuracy the numerically sampled random variable X—plotted in black dots. When σX becomes larger, a moderate deviation is observed, as expected.

Fig. C1.
Fig. C1.

Variance and mean of the square root random variable. The black dots are calculated as follows: for each value of σX, the variance and mean of the random variable X are numerically estimated by taking the square root of 105 draws of a normal random variable X with mean 2 and standard deviation σX. The blue and orange solid lines indicate the truncated predictions of Eq. (C2) for the variance and mean, respectively.

Citation: Journal of the Atmospheric Sciences 81, 1; 10.1175/JAS-D-23-0155.1

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  • Cooper, W. A., 1989: Effects of variable droplet growth histories on droplet size distributions. Part I: Theory. J. Atmos. Sci., 46, 13011311, https://doi.org/10.1175/1520-0469(1989)046<1301:EOVDGH>2.0.CO;2.

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  • Devenish, B. J., K. Furtado, and D. J. Thomson, 2016: Analytical solutions of the supersaturation equation for a warm cloud. J. Atmos. Sci., 73, 34533465, https://doi.org/10.1175/JAS-D-15-0281.1.

    • Search Google Scholar
    • Export Citation
  • Einstein, A., 1905: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys., 322, 549560, https://doi.org/10.1002/andp.19053220806.

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    • Export Citation
  • Eytan, E., A. Khain, M. Pinsky, O. Altaratz, J. Shpund, and I. Koren, 2022: Shallow cumulus properties as captured by adiabatic fraction in high-resolution LES simulations. J. Atmos. Sci., 79, 409428, https://doi.org/10.1175/JAS-D-21-0201.1.

    • Search Google Scholar
    • Export Citation
  • Field, P. R., A. A. Hill, K. Furtado, and A. Korolev, 2014: Mixed-phase clouds in a turbulent environment. Part 2: Analytic treatment. Quart. J. Roy. Meteor. Soc., 140, 870880, https://doi.org/10.1002/qj.2175.

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    • Export Citation
  • Furtado, K., P. R. Field, I. A. Boutle, C. J. Morcrette, and J. M. Wilkinson, 2016: A physically based subgrid parameterization for the production and maintenance of mixed-phase clouds in a general circulation model. J. Atmos. Sci., 73, 279291, https://doi.org/10.1175/JAS-D-15-0021.1.

    • Search Google Scholar
    • Export Citation
  • Gardiner, C., 2009: Stochastic Methods: A Handbook for the Natural and Social Sciences. Springer-Verlag, 447 pp.

  • Grabowski, W. W., and G. C. Abade, 2017: Broadening of cloud droplet spectra through eddy hopping: Turbulent adiabatic parcel simulations. J. Atmos. Sci., 74, 14851493, https://doi.org/10.1175/JAS-D-17-0043.1.

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  • Hoffmann, F., 2020: Effects of entrainment and mixing on the Wegener–Bergeron–Findeisen process. J. Atmos. Sci., 77, 22792296, https://doi.org/10.1175/JAS-D-19-0289.1.

    • Search Google Scholar
    • Export Citation
  • Khain, A. P., and M. Pinsky, 2018: Physical Processes in Clouds and Cloud Modeling. Cambridge University Press, 640 pp., https://doi.org/10.1017/9781139049481.

  • Korolev, A. V., and I. P. Mazin, 2003: Supersaturation of water vapor in clouds. J. Atmos. Sci., 60, 29572974, https://doi.org/10.1175/1520-0469(2003)060<2957:SOWVIC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Korolev, A. V., and P. R. Field, 2008: The effect of dynamics on mixed-phase clouds: Theoretical considerations. J. Atmos. Sci., 65, 6686, https://doi.org/10.1175/2007JAS2355.1.

    • Search Google Scholar
    • Export Citation
  • Korolev, A. V., and Coauthors, 2017: Mixed-phase clouds: Progress and challenges. Ice Formation and Evolution in Clouds and Precipitation: Measurement and Modeling Challenges, Meteor. Monogr., No. 58, Amer. Meteor. Soc., https://doi.org/10.1175/AMSMONOGRAPHS-D-17-0001.1.

  • Li, X.-Y., G. Svensson, A. Brandenburg, and N. E. L. Haugen, 2019: Cloud-droplet growth due to supersaturation fluctuations in stratiform clouds. Atmos. Chem. Phys., 19, 639648, https://doi.org/10.5194/acp-19-639-2019.

    • Search Google Scholar
    • Export Citation
  • Manton, M. J., 1979: On the broadening of a droplet distribution by turbulence near cloud base. Quart. J. Roy. Meteor. Soc., 105, 899914, https://doi.org/10.1002/qj.49710544613.

    • Search Google Scholar
    • Export Citation
  • McGraw, R., and Y. Liu, 2006: Brownian drift-diffusion model for evolution of droplet size distributions in turbulent clouds. Geophys. Res. Lett., 33, L03802, https://doi.org/10.1029/2005GL023545.

    • Search Google Scholar
    • Export Citation
  • Paoli, R., and K. Shariff, 2009: Turbulent condensation of droplets: Direct simulation and a stochastic model. J. Atmos. Sci., 66, 723740, https://doi.org/10.1175/2008JAS2734.1.

    • Search Google Scholar
    • Export Citation
  • Pavliotis, G. A., 2014: Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations. Vol. 60, Springer, 339 pp.

  • Pavliotis, G. A., and A. M. Stuart, 2008: Multiscale Methods: Averaging and Homogenization. Springer, 310 pp.

  • Prabha, T. V., A. Khain, R. S. Maheshkumar, G. Pandithurai, J. R. Kulkarni, M. Konwar, and B. N. Goswami, 2011: Microphysics of premonsoon and monsoon clouds as seen from in situ measurements during the Cloud Aerosol Interaction and Precipitation Enhancement Experiment (CAIPEEX). J. Atmos. Sci., 68, 18821901, https://doi.org/10.1175/2011JAS3707.1.

    • Search Google Scholar
    • Export Citation
  • Prabhakaran, P., A. S. M. Shawon, G. Kinney, S. Thomas, W. Cantrell, and R. A. Shaw, 2020: The role of turbulent fluctuations in aerosol activation and cloud formation. Proc. Natl. Acad. Sci. USA, 117, 16 83116 838, https://doi.org/10.1073/pnas.2006426117.

    • Search Google Scholar
    • Export Citation
  • Pruppacher, H. K., and J. D. Klett, 2010: Microphysics of Clouds and Precipitation. Springer, 954 pp.

  • Risken, H., 1989: The Fokker-Planck Equation: Methods of Solution and Applications. 2nd ed. Springer-Verlag, 472 pp.

  • Rodean, H. C., 1996: Stochastic Lagrangian Models of Turbulent Diffusion. Meteor. Monogr., No. 48, Amer. Meteor. Soc., 84 pp.

  • Rogers, R. R., and M. K. Yau, 1989: A Short Course in Cloud Physics. 3rd ed. Pergamon Press, 304 pp.

  • Saito, I., T. Gotoh, and T. Watanabe, 2019: Broadening of cloud droplet size distributions by condensation in turbulence. J. Meteor. Soc. Japan, 97, 867891, https://doi.org/10.2151/jmsj.2019-049.

    • Search Google Scholar
    • Export Citation
  • Sardina, G., F. Picano, L. Brandt, and R. Caballero, 2015: Continuous growth of droplet size variance due to condensation in turbulent clouds. Phys. Rev. Lett., 115, 184501, https://doi.org/10.1103/PhysRevLett.115.184501.

    • Search Google Scholar
    • Export Citation
  • Shaw, R. A., 2003: Particle-turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech., 35, 183227, https://doi.org/10.1146/annurev.fluid.35.101101.161125.

    • Search Google Scholar
    • Export Citation
  • Siebert, H., and R. A. Shaw, 2017: Supersaturation fluctuations during the early stage of cumulus formation. J. Atmos. Sci., 74, 975988, https://doi.org/10.1175/JAS-D-16-0115.1.

    • Search Google Scholar
    • Export Citation
  • Siewert, C., J. Bec, and G. Krstulovic, 2017: Statistical steady state in turbulent droplet condensation. J. Fluid Mech., 810, 254280, https://doi.org/10.1017/jfm.2016.712.

    • Search Google Scholar
    • Export Citation
  • Squires, P., 1952: The growth of cloud drops by condensation. Aust. J. Sci. Res., 5, 59.

  • Uhlenbeck, G. E., and L. S. Ornstein, 1930: On the theory of the Brownian motion. Phys. Rev., 36, 823841, https://doi.org/10.1103/PhysRev.36.823.

    • Search Google Scholar
    • Export Citation
  • Zelinka, M. D., T. A. Myers, D. T. McCoy, S. Po-Chedley, P. M. Caldwell, P. Ceppi, S. A. Klein, and K. E. Taylor, 2020: Causes of higher climate sensitivity in CMIP6 models. Geophys. Res. Lett., 47, e2019GL085782, https://doi.org/10.1029/2019GL085782.

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    • Export Citation
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  • Abade, G. C., W. W. Grabowski, and H. Pawlowska, 2018: Broadening of cloud droplet spectra through eddy hopping: Turbulent entraining parcel simulations. J. Atmos. Sci., 75, 33653379, https://doi.org/10.1175/JAS-D-18-0078.1.

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  • Cooper, W. A., 1989: Effects of variable droplet growth histories on droplet size distributions. Part I: Theory. J. Atmos. Sci., 46, 13011311, https://doi.org/10.1175/1520-0469(1989)046<1301:EOVDGH>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Devenish, B. J., K. Furtado, and D. J. Thomson, 2016: Analytical solutions of the supersaturation equation for a warm cloud. J. Atmos. Sci., 73, 34533465, https://doi.org/10.1175/JAS-D-15-0281.1.

    • Search Google Scholar
    • Export Citation
  • Einstein, A., 1905: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys., 322, 549560, https://doi.org/10.1002/andp.19053220806.

    • Search Google Scholar
    • Export Citation
  • Eytan, E., A. Khain, M. Pinsky, O. Altaratz, J. Shpund, and I. Koren, 2022: Shallow cumulus properties as captured by adiabatic fraction in high-resolution LES simulations. J. Atmos. Sci., 79, 409428, https://doi.org/10.1175/JAS-D-21-0201.1.

    • Search Google Scholar
    • Export Citation
  • Field, P. R., A. A. Hill, K. Furtado, and A. Korolev, 2014: Mixed-phase clouds in a turbulent environment. Part 2: Analytic treatment. Quart. J. Roy. Meteor. Soc., 140, 870880, https://doi.org/10.1002/qj.2175.

    • Search Google Scholar
    • Export Citation
  • Furtado, K., P. R. Field, I. A. Boutle, C. J. Morcrette, and J. M. Wilkinson, 2016: A physically based subgrid parameterization for the production and maintenance of mixed-phase clouds in a general circulation model. J. Atmos. Sci., 73, 279291, https://doi.org/10.1175/JAS-D-15-0021.1.

    • Search Google Scholar
    • Export Citation
  • Gardiner, C., 2009: Stochastic Methods: A Handbook for the Natural and Social Sciences. Springer-Verlag, 447 pp.

  • Grabowski, W. W., and G. C. Abade, 2017: Broadening of cloud droplet spectra through eddy hopping: Turbulent adiabatic parcel simulations. J. Atmos. Sci., 74, 14851493, https://doi.org/10.1175/JAS-D-17-0043.1.

    • Search Google Scholar
    • Export Citation
  • Hoffmann, F., 2020: Effects of entrainment and mixing on the Wegener–Bergeron–Findeisen process. J. Atmos. Sci., 77, 22792296, https://doi.org/10.1175/JAS-D-19-0289.1.

    • Search Google Scholar
    • Export Citation
  • Khain, A. P., and M. Pinsky, 2018: Physical Processes in Clouds and Cloud Modeling. Cambridge University Press, 640 pp., https://doi.org/10.1017/9781139049481.

  • Korolev, A. V., and I. P. Mazin, 2003: Supersaturation of water vapor in clouds. J. Atmos. Sci., 60, 29572974, https://doi.org/10.1175/1520-0469(2003)060<2957:SOWVIC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Korolev, A. V., and P. R. Field, 2008: The effect of dynamics on mixed-phase clouds: Theoretical considerations. J. Atmos. Sci., 65, 6686, https://doi.org/10.1175/2007JAS2355.1.

    • Search Google Scholar
    • Export Citation
  • Korolev, A. V., and Coauthors, 2017: Mixed-phase clouds: Progress and challenges. Ice Formation and Evolution in Clouds and Precipitation: Measurement and Modeling Challenges, Meteor. Monogr., No. 58, Amer. Meteor. Soc., https://doi.org/10.1175/AMSMONOGRAPHS-D-17-0001.1.

  • Li, X.-Y., G. Svensson, A. Brandenburg, and N. E. L. Haugen, 2019: Cloud-droplet growth due to supersaturation fluctuations in stratiform clouds. Atmos. Chem. Phys., 19, 639648, https://doi.org/10.5194/acp-19-639-2019.

    • Search Google Scholar
    • Export Citation
  • Manton, M. J., 1979: On the broadening of a droplet distribution by turbulence near cloud base. Quart. J. Roy. Meteor. Soc., 105, 899914, https://doi.org/10.1002/qj.49710544613.

    • Search Google Scholar
    • Export Citation
  • McGraw, R., and Y. Liu, 2006: Brownian drift-diffusion model for evolution of droplet size distributions in turbulent clouds. Geophys. Res. Lett., 33, L03802, https://doi.org/10.1029/2005GL023545.

    • Search Google Scholar
    • Export Citation
  • Paoli, R., and K. Shariff, 2009: Turbulent condensation of droplets: Direct simulation and a stochastic model. J. Atmos. Sci., 66, 723740, https://doi.org/10.1175/2008JAS2734.1.

    • Search Google Scholar
    • Export Citation
  • Pavliotis, G. A., 2014: Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations. Vol. 60, Springer, 339 pp.

  • Pavliotis, G. A., and A. M. Stuart, 2008: Multiscale Methods: Averaging and Homogenization. Springer, 310 pp.

  • Prabha, T. V., A. Khain, R. S. Maheshkumar, G. Pandithurai, J. R. Kulkarni, M. Konwar, and B. N. Goswami, 2011: Microphysics of premonsoon and monsoon clouds as seen from in situ measurements during the Cloud Aerosol Interaction and Precipitation Enhancement Experiment (CAIPEEX). J. Atmos. Sci., 68, 18821901, https://doi.org/10.1175/2011JAS3707.1.

    • Search Google Scholar
    • Export Citation
  • Prabhakaran, P., A. S. M. Shawon, G. Kinney, S. Thomas, W. Cantrell, and R. A. Shaw, 2020: The role of turbulent fluctuations in aerosol activation and cloud formation. Proc. Natl. Acad. Sci. USA, 117, 16 83116 838, https://doi.org/10.1073/pnas.2006426117.

    • Search Google Scholar
    • Export Citation
  • Pruppacher, H. K., and J. D. Klett, 2010: Microphysics of Clouds and Precipitation. Springer, 954 pp.

  • Risken, H., 1989: The Fokker-Planck Equation: Methods of Solution and Applications. 2nd ed. Springer-Verlag, 472 pp.

  • Rodean, H. C., 1996: Stochastic Lagrangian Models of Turbulent Diffusion. Meteor. Monogr., No. 48, Amer. Meteor. Soc., 84 pp.

  • Rogers, R. R., and M. K. Yau, 1989: A Short Course in Cloud Physics. 3rd ed. Pergamon Press, 304 pp.

  • Saito, I., T. Gotoh, and T. Watanabe, 2019: Broadening of cloud droplet size distributions by condensation in turbulence. J. Meteor. Soc. Japan, 97, 867891, https://doi.org/10.2151/jmsj.2019-049.

    • Search Google Scholar
    • Export Citation
  • Sardina, G., F. Picano, L. Brandt, and R. Caballero, 2015: Continuous growth of droplet size variance due to condensation in turbulent clouds. Phys. Rev. Lett., 115, 184501, https://doi.org/10.1103/PhysRevLett.115.184501.

    • Search Google Scholar
    • Export Citation
  • Shaw, R. A., 2003: Particle-turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech., 35, 183227, https://doi.org/10.1146/annurev.fluid.35.101101.161125.

    • Search Google Scholar
    • Export Citation
  • Siebert, H., and R. A. Shaw, 2017: Supersaturation fluctuations during the early stage of cumulus formation. J. Atmos. Sci., 74, 975988, https://doi.org/10.1175/JAS-D-16-0115.1.

    • Search Google Scholar
    • Export Citation
  • Siewert, C., J. Bec, and G. Krstulovic, 2017: Statistical steady state in turbulent droplet condensation. J. Fluid Mech., 810, 254280, https://doi.org/10.1017/jfm.2016.712.

    • Search Google Scholar
    • Export Citation
  • Squires, P., 1952: The growth of cloud drops by condensation. Aust. J. Sci. Res., 5, 59.

  • Uhlenbeck, G. E., and L. S. Ornstein, 1930: On the theory of the Brownian motion. Phys. Rev., 36, 823841, https://doi.org/10.1103/PhysRev.36.823.

    • Search Google Scholar
    • Export Citation
  • Zelinka, M. D., T. A. Myers, D. T. McCoy, S. Po-Chedley, P. M. Caldwell, P. Ceppi, S. A. Klein, and K. E. Taylor, 2020: Causes of higher climate sensitivity in CMIP6 models. Geophys. Res. Lett., 47, e2019GL085782, https://doi.org/10.1029/2019GL085782.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Phase relaxation to equilibrium. The blue curve represents the average of 10 ensemble means of 103 noise realizations of Eq. (19) for an icy cloud, with a prescribed mean updraft of w¯=0.2ms1. The gray shading shows plus and minus two standard deviations of the 10 ensembles. The orange curve is the solution of Eq. (19), where noise is vanishing. The green curve is the exponential relaxation time obtained from the linear Eq. (26). The ambient conditions in this numerical experiment are S(0) = 0.5, T = −10°C, p = 50 500 Pa, r¯=5μm, and Ni = 100 L−1.

  • Fig. 2.

    Supersaturation variance as a function of TKE. The blue colors correspond to the variances for the 2D linear system (30) with red-noise updrafts, whereas the black colors correspond to the homogenized 1D Eq. (26), where updrafts become white in time. The dots are obtained by calculating the variance of 106-s time series with a time step of 10−2 s for each value of TKE. The solid lines are the predicted variance using Eqs. (22a) and (35), for the blue and orange curves, respectively. We highlight that when turbulence is less energetic, both estimates become more similar. The ambient conditions in this numerical experiment are SE = 0, w¯=0, T = −10°C, p = 50 500 Pa, r¯=5μm, and Ni = 100 L−1.

  • Fig. 3.

    Comparison between PDFs. The legend indicates the analytically obtained PDFs {fk}k=15. The example ambient conditions for the calculations are SE = 0, w¯=0, T = −5°C, p = 50 500 Pa, r¯=5μm, σr2=106m2, TKE = 5 m2 s−2, and Ni = 100 L−1.

  • Fig. 4.

    (a),(c) Supercooled liquid cloud fraction and (b),(d) mean LWC as a function of TKE. The supercooled liquid cloud fraction and mean LWC are calculated using Eqs. (45) and (47), respectively, against TKE for each analytical supersaturation distribution {fk}k=15 and for two temperature configurations. The red and orange curves, for f2 and f4, respectively, are almost coinciding. In (a) and (b), a temperature of −10°C is considered, while in (c) and (d), a temperature of −5°C is considered. The rest of the ambient conditions are SE = 0, w¯=0, σr2=106m2, p = 50 500 Pa, r¯=5μm, and Ni = 100 L−1.

  • Fig. 5.

    (a),(c) Supercooled liquid cloud fraction and (b),(d) mean LWC as a function of σr. The supercooled liquid cloud fraction and mean LWC are calculated using Eqs. (45) and (47), respectively, against σr for the analytical distributions f1, f4, and f5 and for two temperature configurations. For small values of σr, the functions f4 and f5 are computationally expensive to evaluate so ergodic averages of 106 seconds are taken instead. In (a) and (b), a temperature of −10°C is considered, while in (b) and (d), a temperature of −5°C is considered. The rest of the ambient conditions are SE = 0, w¯=0, TKE = 2.612 m2 s−2, p = 50 500 Pa, r¯=5μm, and Ni = 100 L−1.

  • Fig. C1.

    Variance and mean of the square root random variable. The black dots are calculated as follows: for each value of σX, the variance and mean of the random variable X are numerically estimated by taking the square root of 105 draws of a normal random variable X with mean 2 and standard deviation σX. The blue and orange solid lines indicate the truncated predictions of Eq. (C2) for the variance and mean, respectively.

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